Where Portfolio Selection sits

Markowitz  (1952) is in effect a connection made between a piece of new computer science (linear programming and techniques such as simplex, and generally constrained optimisation solutions which arose out of the second world war) and an application in financial theory.  He tells the admirably random story of how he was waiting to see his professor when he struck up a conversation with another guy in the room, waiting to see the same professor, the guy being a broker, who suggested to Markowitz that he should apply his computer science algorithms skill to solving finance problems.

And given this random inspiration, he later finds himself in a library reading a book by John Burr Williams and he has a moment of revelation, namely that when you consider portfolios, the expected return on the portfolio is homogeneously just the weighted average of the expected returns of the component securities and so if this was the only criterion which mattered, your portfolio would just be 100% made up of that single portfolio which had the highest expected return.  You might call this the ancestral 'absolute alpha' strategy.  In knowing this single criterion was silly, he drew upon his liberal arts background, his knowledge of the Merchant of Venice, Act 1 Scene 1, as well as his understanding of game theory, particularly the idea of an iterated game and the principle of diversification, to seek out variance as an operational definition of risk.

He now had two dimensions to optimise, maximise returns whilst simultaneously minimise variance.  And finally, when he looks at how portfolio variance is calculated, he has his second moment of inspiration, since this is not just a naive sum of constituent variances, no, the portfolio variance calculation is a different beast.  This feeling, that the behaviour of the atoms are not of the same quality as the behaviour of the mass, is perhaps also what led John Maynard Keynes to posit a macro-economics which was different in quality to the micro- or classical economics of his education.

With normalised security quantities $x_i$ the portfolio variance is $\sum_i \sum_j x_i x_j \sigma_{i,j}$.

His third great moment was in realising that this was a soluble optimisation program, soluble in the case of two or three securities geometrically, but soluble in the general case with linear programming.  Linear programming also allowed for linear constraints to be added, indeed demanded that some be the case; for example that full investment occur, $\sum_i x_i = 1$, and that you can't short, $\forall i, x_i>0$.

However, notice the tension.  We humans often tend to favour one end of the normal distribution over another whereas mathematics doesn't care.  Take the distribution of returns, we cherish, desire even, the right hand side of the returns distribution and fear the left hand side.  So maximising the return on a portfolio makes good sense to us, but variance is not left or right handed.  Minimising variance is minimising the positive semi-variance and minimising the negative semi-variance too.  This is, so to speak, sub-optimal.  We want to avoid downside variance, but we probably feel a lot more positively disposed to upside variance.  Yet the mathematics of variance is side-neutral, yet we plug straight into that maths.